Suppose that we wanted to ask what the most famous equation in physics is. There
are, I imagine, a few candidates.

Some might point to the pre-Newton experimentally-derived “laws” that
people learn about in school, such as Hooke’s Law, Boyle’s law for gases;
maybe Snell’s law for refraction.

Newton’s second law of motion, F = ma, has to be a candidate.

As does Newton’s law of gravity,

Then we have the first law of thermodynamics, which might not be familiar
in the standard form dE = dQ - dW (E the internal energy, Q the
heat supplied by the system and W the work done by the system on
its surroundings), but is basically just a statement of the conservation of
energy, which is very familiar.

I would hope that some people would cite Maxwell’s equations and the
Lorentz force Law, arguably the most important results of classical physics.

Then we have the Schrödinger equation

whose name many people will know even if they don’t know anything else
about quantum mechanics.

The principle of least action and Hamilton’s equations also ought to get short-listed.

But ask a random person on the street to name an equation from physics, and if I were a
gambling man (which I am not) I would be willing to bet that the majority of
them would say E = mc2, even if they don’t understand what the equation
means. So what does it mean?

First of all, we need to ask what the symbols mean. E represents energy, m
represents mass and c the speed of light. Well, that’s simple enough. It means that
mass and energy are equivalent, doesn’t it?

Well, no it doesn’t. In this post, I intend to briefly show where Einstein's equation comes from,
and by doing so take a look at some aspects of its significance.

Before we begin, we need to ask what the speed of light, mass and energy and mass represent in classical physics.

The speed of light is the speed at which light travels in a vacuum. Simple enough.
It is also linked to the strength of the electric and magnetic forces. It plays a particularly important role in special relativity. The
Lorentz transformations which are used to convert the co-ordinates in one invariant reference frame to another are dependent on the relative velocity divided by
the speed of light. In this way, the speed of light is a natural way to convert between durations in time and distances (which we need to do
when considering what the path of a particle is as observed by someone moving at a constant velocity to ourselves).

Mass is an inherent property of matter that does two jobs:

It represents the resistance of an object to accelerating (Newton’s second
law is the definition of this sense of mass).

The gravitational force between two objects is proportional to the product
of their masses (Newton’s law of gravity). It measures both the strength of
the gravitational field generated by the object, and the object’s response
to a gravitational field generated by something else.

You might ask why the mass plays two seemingly independent roles in classical physics:
and if you do ask that, think about it in the right way, and know enough about
geometry, then you are well on the way to uncovering the general theory of relativity.
But I digress.

Energy is a bit different. In Newtonian mechanics, energy is basically
something that you get by throwing together various numbers. It’s main use is
that it is a conserved quantity. Newton’s second and third laws together
mean that the total energy does not change in time. Energy is transferred
from one object to another, but the sum of the energy remains the same. In
fact, it is often easier to deduce the dynamics of a collision by using the
conservation of energy and momentum than it is to explicitly solve Newton’s
equations.

Before giving the equation for the energy, I need a couple of other definitions.
Firstly, we have the momentum, p, which is mass times velocity. This features in
Newton’s second and third equations; Newton’s third equation states that
momentum is also a conserved quantity. Next, the electrostatic potential, which I will
write as cA0. The electric force is the rate of change of the potential with respect to
distance. Then we have the magnetic potential, A, which can be used to compute the
force from magnetism.

Newton’s equation for the energy for a single particle of electric charge q in an
electromagnetic field (excluding gravity and the other forces: gravity doesn’t affect
the conclusions of argument but makes it considerably more complicated if we want
to do gravity in a manner consistent with relativity; the other forces arise as a
result of quantum physics and we are here discussing classical physics) is
then

The first term is known as the kinetic energy, and the second the potential energy.
Because energy is conserved, we can increase the kinetic energy (make the particle go
faster) by reducing the potential energy, or we can make the particle slower and
increase the potential energy, but the sum of the two terms must remain constant. If
we have more than one particle, we can add together their individual energies, and
can transfer kinetic or potential energy from one particle to another, but the total
energy must remain constant.

Einstein realised that this equation was inconsistent with Maxwell’s equations for
electromagnetism. Newton’s equation for the energy is wrong. Einstein also worked
out what the right equation is, and it is usually expressed as (in a form that makes clear the symmetry between energy and momentum)

We can rearrange it to give,

So once again, we seem to have two terms: the potential energy, and something else.
We can transfer energy from the something else to the potential energy and
vice versa, and from one particle to another, but again energy is conserved.
The potential energy has exactly the same form as in Newton’s formula,
but the other term is a bit different from Newton’s formula to the Kinetic
energy.

To find the connection between Einstein’s and the Newtonian expressions, we
need to do a little bit of algebra. First of all, we re-arrange it

Then we do a low momentum expansion,

So we have three terms: the potential energy; one which depends on the momentum,
and whose most significant contribution when the momentum is much smaller than the mass times the speed of light is
the Newtonian kinetic energy (the remaining contributions are much smaller than the experimental precision of the observations that were used to support Newton's formula; but have subsequently been verified by more precise experiments); and a third term, the familiar mc2, which is entirely absent in Newtonian
mechanics. Since in Newtonian mechanics all that matters is energy difference rather than the absolute energy, no pre-twentieth century experiment was affected by its presence.

What this equation tells us is that mass is a type of energy. It is not equivalent to
energy, because it is not the only type of energy (no more than potential energy is
equivalent to energy as a whole). It also suggests that, just as we convert potential
energy into kinetic energy and vice versa, so we can also convert kinetic energy into
mass and vice versa. (One might also think that one can convert potential energy
into mass; but when we get into relativistic quantum physics, the electromagnetic
potential is regarded in a different way, so this becomes a bit more complicated. For
this discussion, I’ll just consider the conversion between kinetic energy and
mass.)

But this last point is troubling philosophically, at least for certain philosophies,
most especially that adopted by Newton. Mass is an intrinsic property of matter. To
create new mass means creating new matter. For example, each electron has the same
mass; the only way we can convert kinetic energy into electron mass is to
create new electrons. This suggests that it is possible to create (and destroy)
matter.

But in the mechanical philosophy – both classical mechanics and quantum
mechanics – the fundamental type of matter, the corpuscles, are supposed to be indestructible (and, the current best
theory suggests that electrons are elementary; though even if they are
composite the same considerations would apply to whatever electrons are made
of). This is one of the primary axioms of
mechanism. The mechanical philosophy supposes that physics reduces to the
different locomotions and arrangements of corpuscles; and has no capacity to
appreciate their creation and destruction. Neither Newton’s equations of
motion nor Schrödinger’s equation can describe the creation or destruction of
matter.

Now, we could perhaps rescue mechanism by saying that conversion between
kinetic and mass energy never occurs in practice; there is nothing in Einstein’s
equation saying that mandates that it must happen. It just suggests that it might be
possible. The only requirement of the equation is that mass energy plus kinetic
energy plus potential energy remains constant. Unfortunately, experiment now
intervenes and tells us that kinetic energy is indeed converted into new particles. So
the natural implication of Einstein’s equation was right, and the original versions of
the mechanical philosophy were wrong.

Interestingly, though, Aristotle was quite happy with one form of matter being
converted into another; for example when wood burns and becomes fire.
This example is, of course, taken from Greek physics, which we know is
problematic. But even if the reasons why Aristotle thought that his philosophy
was necessary were false, it does not mean that his philosophy was in fact
false. It just means that the arguments he used to support it were bad. One can defend a good idea through bad argumentation.
Aristotle's philosophy becomes relevant again when we find a better example of what he was proposing; and thus his conclusion that the correct metaphysics must be general enough to admit to the possibility of transmutation from one type of matter to another. Modern physics comes to a similar conclusion: that
one form of matter can emit, absorb or transform into another. The only
requirement that Einstein left us with was that energy has to be conserved.
New matter arises from the kinetic energy of something else. You cannot
create from nothing, but only by taking from something else. Moreover, that
something else has to have enough kinetic energy or momentum to compensate
for the mass of the new particles; the effect has to be contained in some
way within the cause. So find ourselves with two fundamental Aristotelian
principles: the principle of causality (every actualised potential is caused by
something already actual), and the principle of proportionate causality (an effect
can only come about if the cause possessed the power to bring about that
effect; i.e. the effect is allowed as a possibility by the inherent nature of the
cause).

So what we need to do is construct a new physics that contains a mathematical
description of material particles being created and destroyed and has the principles of causality and proportionate causality built in.

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